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Theorem eldmcoss 38156
Description: Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.)
Assertion
Ref Expression
eldmcoss (𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Distinct variable groups:   𝑢,𝐴   𝑢,𝑅   𝑢,𝑉

Proof of Theorem eldmcoss
StepHypRef Expression
1 dmcoss3 38151 . . 3 dom ≀ 𝑅 = dom 𝑅
21eleq2i 2818 . 2 (𝐴 ∈ dom ≀ 𝑅𝐴 ∈ dom 𝑅)
3 eldmcnv 38043 . 2 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
42, 3bitrid 282 1 (𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wex 1774  wcel 2099   class class class wbr 5153  ccnv 5681  dom cdm 5682  ccoss 37876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-coss 38109
This theorem is referenced by:  eldmcoss2  38157  eldm1cossres  38158
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